Topics in Applied Analysis and Optimisation (eBook)

Preface 6
Contents 9
Recent Trends and Views on Elliptic Quasi-Variational Inequalities 16
Amal Alphonse, Michael Hintermüller, and Carlos N. Rautenberg 16
1 Introduction 16
1.1 The basic setting and problem formulation 18
2 Some existence theory 19
2.1 Compactness and Mosco convergence 19
2.1.1 The result of Boccardo and Murat 21
2.1.2 Gradient and further cases 21
2.2 Order approaches 22
3 Solution methods and algorithms 24
3.1 Contraction results for T 24
3.1.1 Obstacle case 25
3.1.2 Gradient and further cases 25
3.2 The map KPK and extensions to Lions–Stampacchia 26
3.3 Order approaches: solution methods for m(f) and M(f) 29
3.4 Regularization methods 31
3.4.1 Extended Moreau–Yosida and Semismooth Newton 31
3.5 Gerhardt-type regularization for the gradient case 34
3.6 Drawbacks of the iteration yn+1=T(yn) 35
4 Optimal control problems 36
5 Differentiability 37
5.1 Directional differentiability for QVIs 39
6 Conclusion 43
References 43
The Incompatibility Operator: from Riemann's Intrinsic View of Geometry to a New Model of Elasto-Plasticity 47
Samuel Amstutz, Nicolas Van Goethem 47
1 On the origin of curvature in science and the birth of intrinsic views 48
2 Curvature in nonlinear elasticity 49
3 Incompatibility in linearized elasticity and path integral formulae 51
4 The legacy of Ekkehart Kröner: the geometry of a crystal with dislocations 53
4.1 The geometric approach at the macroscale 53
4.2 Parallel displacement and curvature 55
4.3 The non-Riemannian crystal manifold 57
4.4 Internal and external observers 58
4.5 Inelastic effects and notion of eigenstrain 59
5 A geometric conception of linearized elasticity: the intrinsic approach 59
5.1 Gauss vs. Riemann in linearized elasticity 59
5.2 Ciarlet's intrinsic approach to linearized elasticity 60
6 The classical route to plasticity 62
6.1 The mathematical approaches: two perspectives 62
6.2 Conventional (0th-order) elasto-plasticity models 64
7 Gradient elasto-plasticity for continua with dislocations: towards an incompatibility-driven model 65
7.1 The size effect 65
7.2 Gradient models 66
7.3 Our approach: a gradient model based on the strain incompatibility 66
7.4 Link with classical elasto-plasticity models 68
8 The incompatibility operator: functional framework 69
8.1 Divergence-free lifting, Green formula and applications 69
8.2 Saint-Venant compatibility conditions and Beltrami decomposition 72
8.3 Orthogonal decompositions 72
8.4 Boundary value problems for the incompatibility 73
9 Towards an intrinsic approach to linearized elasto-plasticity 74
9.1 Objectivity and principle of virtual powers 74
9.2 Constitutive law 76
9.3 Equilibrium equations 76
9.4 Interpretation of the external power and kinematical framework 77
9.5 Existence results and elastic limit 77
9.6 Example: bar in traction 78
9.7 Incremental formulation of hardening problems 80
Acknowledgements 81
References 81
Nonlocal Phase Field Models of Viscous Cahn–Hilliard Type 85
Pierluigi Colli, Gianni Gilardi, Jürgen Sprekels 85
1 Introduction 85
1.1 About the model and related problems 86
1.2 Nonlocal operators 88
1.3 Overview of some related contribution 90
1.4 About well-posedness and regularity results 91
1.5 The optimal control problem for a logarithmic potential 91
1.6 Commenting on the optimal control problem 92
1.7 The optimal control problem in the double-obstacle case 93
1.8 The deep quench limit procedure 94
2 Results 95
2.1 The mathematical framework 96
2.2 Mathematical problem and general results 97
2.3 Special case for the existence result 100
2.4 Directional differentiability of the control-to-state mapping 104
2.5 Existence and first-order necessary conditions of optimality 107
2.6 The double-obstacle case 110
Acknowledgments 111
References 111
Invariant and Quasi-invariant Measures for Equations in Hydrodynamics 115
Ana Bela Cruzeiro and Alexandra Symeonides 115
1 Introduction 115
2 The two-dimensional Euler equation (periodic case) 116
2.1 Formulation of the equation 116
2.2 Invariant quantities and Gibbs measures 118
2.3 The vorticity flow 119
2.4 Lagrangian flows 122
3 The two-dimensional Euler equation (non periodic case) 122
4 A modified Euler equation 123
4.1 The vorticity flow for the modified Euler equation 124
4.2 Quasi-invariant measures 125
4.3 Statistical solutions 126
5 The two-dimensional averaged-Euler equations 128
5.1 Formulation of the equations and invariant quantities 128
5.2 Gaussian invariant measures and statistical solutions 129
5.3 Surface measures 131
References 133
Long-range Phase Coexistence Models: Recent Progress on the Fractional Allen-Cahn Equation 135
Serena Dipierro and Enrico Valdinoci 135
1 Prelude 135
2 The classical Allen-Cahn equation 136
3 The fractional Allen-Cahn equation 140
Closing remarks 149
Acknowledgements 149
References 150
Elements of Statistical Inference in 2-Wasserstein Space 153
Johannes Ebert, Vladimir Spokoiny and Alexandra Suvorikova 153
1 Introduction 153
2 Monge-Kantorovich distance for location-scatter family 157
3 Bootstrap procedure for confidence sets 161
4 Application to change point detection 162
5 Algorithms 164
6 Experiments 164
6.1 Coverage probability of the true object * 164
6.2 Experiments on the real data 167
6.3 Application to change point detection 168
References 171
On the Use of ADMM for Imaging Inverse Problems: the Pros and Cons of Matrix Inversions 173
Mário A. T. Figueiredo 173
1 Introduction 173
2 General Problem Formulation 175
3 The Alternating Direction Method of Multipliers 176
3.1 The Standard ADMM 176
3.2 Using ADMM for More than Two Functions 178
4 Linear Observations with Gaussian Noise 180
4.1 Observation Model 180
4.2 Tikhonov Analysis Regularization 180
4.2.1 Periodic Deconvolution 181
4.2.2 Deconvolution with Unknown Boundaries 182
4.2.3 Image Inpainting 184
4.2.4 Compressive Fourier Imaging 184
4.3 Tikhonov Synthesis Regularization 184
4.3.1 Periodic Deconvolution 185
4.3.2 Deconvolution with Unknown Boundaries 185
4.3.3 Image Inpainting 186
4.3.4 Compressive Fourier Imaging 186
4.4 Morozov Analysis Regularization 186
5 Poissonian Observations 188
5.1 Observation Model 188
5.2 Tikhonov Analysis and Synthesis Regularization 188
6 Hybrid Analysis-Synthesis Regularization 190
7 Conclusions and Discussion 191
Acknowledgments 192
References 192
Models and Numerical Methods for Electrolyte Flows 196
Jürgen Fuhrmann, Clemens Guhlke, Alexander Linke, Christian Merdon and Rüdiger Müller 196
1 Introduction 196
2 Continuum models 197
2.1 Bulk equations 198
2.2 Reformulation in species activities 200
2.3 Analytical treatment 201
3 Numerical methods 202
3.1 Previous work 202
3.2 Thermodynamically consistent finite volume methods for species transport 203
3.3 Pressure robust, divergence free finite elements for fluid flow. 205
3.4 Coupling strategy. 207
4 Numerical examples 207
4.1 Infinite pore with charged walls 208
4.2 Slit between two infinite plates 209
4.3 Finite pore with charged walls 211
4.4 Spurious velocities in electrophoresis 213
4.5 Ionic current rectification and flow vortex in a conical nanopore 215
5 Conclusions 218
Acknowledgement 218
References 218
Consequences of Uncertain Friction for the Transport of Natural Gas through Passive Networks of Pipelines 223
Holger Heitsch and Nikolai Strogies 223
1 Introduction 223
2 Results for the state equations 226
2.1 Steady states 226
Special case of tree networks 227
2.2 Time dependent problems 228
2.3 Extension to passive networks 231
3 Uncertainty quantification for the semilinear model 233
4 Numerical Realization 235
4.1 Discussion of numerical examples 237
Example 1 237
Example 2 241
5 Determining probabilities of feasibility sets 243
5.1 Spheric-radial decomposition 244
5.2 Preliminary numerical results 246
Example 1 246
Example 2 248
References 249
Probabilistic Methods for Spatial Multihop Communication Systems 251
Benedikt Jahnel and Wolfgang König 251
1 Introduction 251
2 Basics of the mathematical modeling 254
2.1 Location of users: The Poisson point process 254
2.2 Connectivity: Continuum percolation 254
2.3 Palm calculus 255
2.4 Interference 256
2.5 Large deviations 258
3 Continuum percolation in random environments 260
3.1 Main example: Voronoi tessellation 260
3.2 The critical user intensity for percolation 261
3.3 Asymptotics for the percolation probability 263
3.3.1 Large communication radius 263
3.3.2 Large user intensity 264
3.3.3 Coupled limits 264
Summary 265
4 Large deviations in high-density networks with interference and capacity constraints 266
Interference constraints 267
Capacity constraints 270
5 Random message routing 273
5.1 The model and its motivation 273
5.2 A law of large numbers for the message flow in the high-density limit 275
5.3 Analytic properties of the message trajectory flow 277
5.3.1 Large distances, many hops 277
5.3.2 High interference punishment 278
5.4 Illustrations 278
References 279
Mathematical Modeling of Semiconductors: From Quantum Mechanics to Devices 281
Markus Kantner, Alexander Mielke, Markus Mittnenzweig, and Nella Rotundo 281
1 Introduction 281
2 The van Roosbroeck system 285
2.1 Carrier flux densities and chemical potentials 285
2.2 Boundary conditions 287
3 Mathematical modeling based on thermodynamical principles 288
3.1 The GENERIC framework 288
3.2 Damped Hamiltonian systems 290
3.3 Additive structure of dissipative contributions 290
3.4 Dissipative coupling between different components 291
4 Semiconductor modeling via damped Hamiltonian systems 292
4.1 The state variables and free energy 292
4.2 The van Roosbroeck system as gradient system 293
4.3 Reactions between and transport of charge carriers 294
4.4 Gradient structure for general carrier statistics 295
4.5 Dissipative quantum mechanics 296
4.6 The van Roosbroeck system coupled to a quantum system 298
4.7 Quantum-classical coupling via Onsager operators 301
4.8 Further dissipative coupling strategies 303
References 304
Gradient Structures for Flows of Concentrated Suspensions 306
Dirk Peschka, Marita Thomas, Tobias Ahnert, Andreas Münch, Barbara Wagner 306
1 Introduction 306
2 Model for a concentrated suspension 309
3 Gradient flow for two-phase flows of concentrated suspensions 311
3.1 Notation and states 313
3.2 The triple (V,R,E) for flows of concentrated suspensions 317
3.3 PDE system obtained by the gradient flow formulation 325
4 Conclusion 327
References 328
Variational and Quasi-Variational Inequalities with Gradient Type Constraints 330
José Francisco Rodrigues and Lisa Santos 330
1 Introduction 330
2 Stationary problems 333
2.1 A general p-framework 333
2.2 Well-posedness of the variational inequality 334
2.3 Lagrange multipliers 338
2.4 The quasi-variational solution via compactness 341
2.5 The quasi-variational solution via contraction 343
2.6 Applications 345
3 Evolutionary problems 348
3.1 The variational inequality 348
3.2 Equivalent formulations when L= 353
3.3 The scalar quasi-variational inequality with gradient constraint 355
3.4 The quasi-variational inequality via compactness and monotonicity 358
3.5 The quasi-variational solution via contraction 361
3.6 Applications 365
References 368
Models of Dynamic Damage and Phase-field Fracture, and their Various Time Discretisations 373
Tomáš Roubícek 373
1 Introduction 373
2 Models of damage at small strains 374
3 Phase-field concept towards fracture 385
4 Various time discretisations 388
4.1 Implicit ``monolithic'' discretisation in time 388
4.2 Fractional-step (staggered) discretisation 392
4.3 Explicit time discretisation blackoutlined 394
5 Concluding remarks – some modifications 397
5.1 Combination with creep or plasticity 397
5.2 Damage models at large strains 400
References 403